suppose we have a grade list: $ \text{grades}=\{2,3,5,7,8,10,9,9.75,8,0,11,10,10,3,5.25,13,14,20,18,9\}; $
which mean equals to 8.75 and Standard deviation is 5.06471. we want to improve the grade average using normal distribution and increasing student grades. any idea how?
Here's one possible answer:
I assume these grades are out of $20$. As one student has attained the maximum, we can't just shift all grades up by some amount, as that grade of $20$ among others might exceed the maximum.
Note that $20$ is some number of standard deviations above the norm, $Z(20) = (20 - \mu_{old})/\sigma_{old}$.
Suppose we shift all grades up by $S$ and hence the new mean is $$\mu_{new} = \mu_{old} + S$$ Then we want $20$ to remain fixed at $20$. We do that by modifying down the standard deviation to $\sigma_{new}$ such that $Z(20)$ remains constant, i.e., choose $\sigma_{new}$ such that
$$Z(20) = \frac{20 - \mu_{new}}{\sigma_{new}} $$
That is $\displaystyle \sigma_{new} = \frac{20 - \mu_{new}}{Z(20)}$.
Each old grade $g$ has its own $Z$ score, $Z(g) = (g - \mu_{old})/\sigma_{old}$. The rule then for rescaling a grade $g$ is $$g \mapsto \mu_{new} + Z(g)\sigma_{new}$$
Checking, the maximum score of $20$ is fixed:
$$20 \mapsto \mu_{new} + Z(20) \cdot \frac{20 - \mu_{new}}{Z(20)} = 20$$
For example, if the mean increased by $3$, here's how applying the rule would modify the scores: